On small traveling waves to the mass critical fractional NLS

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Abstract

We consider the mass critical fractional (NLS)
$$\begin{aligned} i\partial _{t}u-\left| D\right| ^{s}u+u\left| u\right| ^{2s}=0,\text { }x\in \mathbb {R},\text { }1<s<2. \end{aligned}$$
We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in Gérard et al (A two soliton with transient turbulent regime for the one dimensional cubic half wave, 2018) for the critical case \(s=1\), but with a completely different asymptotic profile when the mass vanishes.

Mathematics Subject Classification

35Q55 35B40 

Notes

Acknowledgements

Both authors are supported by the ERC-2014-CoG 646650 SingWave. P.R. would like to thank A. Soffer for stimulating discussions about this work and the Central China Normal University, Wuhan, where part of this work was done.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire J.A. DieudonnéUniversité de la Côte d’AzurNiceFrance

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